Electric–magnetic duality and charge lattices

Electric–magnetic duality is the symmetry of Maxwell’s equations under exchange of the electric and magnetic fields: exact in empty space, and extended to matter only if magnetic charges exist alongside electric ones. A particle with magnetic charge alone is a magnetic monopole, a north pole without a south pole; none has ever been detected, and the theory of monopoles and of the dualities that relate them to ordinary charges is established physics. The Broken-Born program meets this subject three times. One of its founding ideas tried to explain the absence of monopoles and died; a weaker form of the same idea survives; and the program observes that choosing which particles count as electric is the same mathematical operation as choosing which coordinates count as position. Whether those two choices are one structure is an open conjecture. A claim found only in a popular article, that the program’s second matter sector is the magnetic side of our electromagnetism, is recorded here as dead.

Stands: magnetic monopoles, the modular group SL(2, ℤ), S-duality and Montonen–Olive, the dual superconductor and the magnetic mirror-family, as established literature; conjugate monopoles as ontology [D] and the symplectic charge lattice [D — structural; math T], as program claims · Died: conjugate monopoles as the explanation of the null result, the magnetic identification (popsci-only) · Open: one modular structure for both polarizations (OP8), with its folded electric-sublattice clause · Popsci-only: besides the dead identification, #12’s conditional “inside every proton”, folded on the dual-superconductor page

Map#

The section follows #11 §11, “Polarized charge lattices: monopoles, duality frames, and the recurring matrix”, which first appeared in the checkpoint’s v2.0. Literature comes first, then the program’s three claims in the order of §11.1–§11.3, then the popular article’s claim. The number 34 appears on several pages with two different meanings: the magnetic coupling of our own electromagnetism, which is established physics (#11 §11.2), and the coupling α_d ≈ 34 of the retired soliton candidate’s hidden monopoles, which is dead. The pages keep them apart.

  • Magnetic monopoles comes first. Its six pages hold Dirac quantization and the Dirac–Schwinger–Zwanziger (DSZ) pairing of charges, the ’t Hooft–Polyakov monopole of a broken gauge symmetry, the Witten effect, the electric charge a vacuum angle gives such a monopole, the Kibble mechanism by which a cooling universe makes such defects, the monopole problem, and the observational flux bounds. All six are literature, and every monopole claim the program has made answers to them.
  • The modular group SL(2, ℤ) is the algebra the dualities share: the 2 × 2 whole-number matrices of determinant 1, generated by S: z → −1/z and T: z → z + 1 and acting on the upper half-plane. It is the symmetry group of a lattice of electric and magnetic charges, with Sp(2r, ℤ) for several gauge fields, and the duality group of the complexified coupling τ_c, in which S is exact only where S-duality is. The page sets out the matrix of S that #11 finds again in Born reciprocity and in the modular inversion Z → −Z⁻¹, where it belongs to a continuous group rather than this discrete one, and the program’s wider use of the word “modular”.
  • S-duality and Montonen–Olive is the quantum form of electric–magnetic duality: Maxwell’s exchange of E and B as a quarter-turn, the complexified coupling τ_c and the generator S: τ_c → −1/τ_c, the Montonen–Olive observation that g → 4π/g swaps the W bosons and the monopoles, the spin mismatch that confines exact duality to supersymmetric theories, Seiberg’s magnetic quarks, and T-, S- and U-duality in string theory. It also records the exactness burden that round 2 placed on the soliton manuscript, and how worklog item R1, Paul’s decision on F2 and #14 v3 discharged it.
  • The dual superconductor is the leading picture of quark confinement: why quarks are never free, how a superconductor squeezes magnetic flux into tubes, and how a condensate of magnetic monopoles would do the same to the color-electric field. #12’s conditional “inside every proton” is folded in as popsci-only.
  • The magnetic mirror-family collects these results as the fate of P.R.’s founding supposition 11, “alive as established physics”. It explains why Dirac’s condition makes the mirror heavy and bound, quotes #11’s “4π/e² ≈ 34” with the factor-of-four convention spread of worklog item R2, points to the flag on the expression (flag 28 in review/wiki-build-flags.md), separates this 34 from the dead α_d ≈ 34, and sets out the parallel #11 draws with the hidden second metric of §7, the momentum graviton.
  • Obituary: conjugate monopoles explain the null result is the death of the explanatory half of founding supposition 5, by argument in #11 §11.1: the relevant duality is electric–magnetic, not Born reciprocity, and inflation already accounts for the absence of monopoles [D, T]. It traces the claim through row 8 of the lab notebook and v1, records the “empty set” and “measure zero” wordings of the split, and lists what survives.
  • Conjugate monopoles as ontology is the half that survived [D]: if the momentum-polarized sector has monopoles or other topological solitons, P4 makes them gravitate and gives them no other coupling to our matter. The page records the dark-matter candidacy the ontology fed, from v2.0 through the handoff to its retirement, the Weyl-semimetal monopoles in momentum space as the laboratory shadow, and the untested topological loophole.
  • The symplectic charge lattice is the program’s structural observation [D — structural; math T]: the DSZ pairing makes charge space symplectic, mutually local charges form a Lagrangian sublattice, a duality frame is a Lagrangian polarization, and one matrix generates Born’s ρ, the modular inversion Z → −Z⁻¹ and S-duality. It compares charge space with phase space, gives the d = 1 checks of the phase-space side in verify/toy_coset.py, and records the critiques made in rounds 2 and 3 of the soliton manuscript’s review.
  • Open: one modular structure for both polarizations (OP8) is the conjecture [O] that the phase-space and charge-space polarizations are components of one enlarged structure, on the model of T- and S-duality inside U-duality. It records #14 v2’s stronger identification and why it was set aside, the unaddressed demand that the order parameter also select the electric charges, and the item’s absence from the current queue.
  • Obituary: the magnetic identification (popsci-only) comes last. It records a claim found only in the popular article #12, that the momentum-polarized sector is the magnetic side of electromagnetism with a coupling “about 34, against our 1/137”. It was never a program claim, and it conflicts with the dead α_d ≈ 34, with strict sector separation and with row 7.

Boundaries#

Backward Gravitational waves from phase transitions Magnetic monopoles Forward